INADMISSIBILITY AND ADMISSIBILITY RESULTS FOR UNBIASED LOSS ESTIMATORS BASED ON GAUSS-MARKOV ESTIMATORS  

作　　者：吴启光 

机构地区：[1]Institute of Systems Science, Academia Sinica, Beijing 100080, China

出　　处：《Acta Mathematicae Applicatae Sinica》1993年第3期281-288,共8页应用数学学报（英文版）

基　　金：This project is supported by the National Natural Science Foundation of China

摘　　要：Let Y be distributed according to an n-variate normal distribution with a mean Xβ and a nonsingular covariance matrix σ~2V,where both X and V are known,β∈R^p is a parameter,σ>0 is known or unknown.Denote β=(X'V-~1X)-X'V-~1Y and S^2=(Y-Xβ)'V^(-1)(Y-Xβ).Assume that Eβ is linearly estimable.When σ is known,it is proved that the unbiased loss estimator σ~2tr(F(X'V-~1X)-F')of(Fβ-Fβ)'(Fβ-Fβ)is admissible for rank (F)=k≤4 and inadmissible for k≥5 with the squared error loss[a-(Fβ-Fβ)'(Fβ-Fβ)]~2 When σ is unknown and rank (X)<n,it is established that the loss estimator cS^2,where c is any nonnegative constant,of(Fβ-Fβ)'(Fβ-Fβ)is inadmissible and that the unbiased loss estimator tr(F(X'V^(-1)X)-F')ofσ^(-2)(Fβ-Fβ)'(Fβ-Fβ)is admissible for k≤4,and inadmissible for k≥5 with squared error loss.Let Y be distributed according to an n-variate normal distribution with a mean Xβ and a nonsingular covariance matrix σ~2V,where both X and V are known,β∈R^p is a parameter,σ>0 is known or unknown.Denote β=(X'V-~1X)-X'V-~1Y and S^2=(Y-Xβ)'V^(-1)(Y-Xβ).Assume that Eβ is linearly estimable.When σ is known,it is proved that the unbiased loss estimator σ~2tr(F(X'V-~1X)-F')of(Fβ-Fβ)'(Fβ-Fβ)is admissible for rank (F)=k≤4 and inadmissible for k≥5 with the squared error loss[a-(Fβ-Fβ)'(Fβ-Fβ)]~2 When σ is unknown and rank (X)<n,it is established that the loss estimator cS^2,where c is any nonnegative constant,of(Fβ-Fβ)'(Fβ-Fβ)is inadmissible and that the unbiased loss estimator tr(F(X'V^(-1)X)-F')ofσ^(-2)(Fβ-Fβ)'(Fβ-Fβ)is admissible for k≤4,and inadmissible for k≥5 with squared error loss.

分 类 号：O1[理学—数学]